Probabilities are the study of "chance". When we calculate the probability of something occurring we are calculating the likelihood of it happening.
Studying probabilities will allow us to answer questions like:
Before answering such questions we need to learn some terminology related to probabilities, this is simply to know how to express ourselves in this topic.
In probabilities, an experiment is a process (could be "anything") in which there are one or more (usually more) possible outcomes each of which depends on chance.
Here we show a couple of examples that show an experiment along with their outcomes.
Given an experiment, the sample space consists of all the possible outcomes of the experiment.
The sample space is usually written, or illustrated, using one of the following:
Write the sample space for each of the experiments, listed below.
Given an experiment, along with its possible outcomes, an event is the name given to either one of the possible outcomes, or a group of outcomes.
Events are usually referred to using a capital letter, such as \(A\), \(B\), \(C\), ... .
The following show a couple of examples of single events.
At times we'll be interested in the comination of two, or more, single events.
For instance, given two events \(A\) and \(B\) we may be interested in the event "\(A\) and \(B\)" or the event "\(A\) or \(B\)"
When picking a card at random from a deck of \(52\) playing cards, we may be interested in the likelihood of picking a card that is both: even and red.
In this case we could define both events:
A probability is a number expressed as either:
Given an event \(A\), the probability of event \(A\) occurring is written: \[p\begin{pmatrix}A \end{pmatrix}\] Read: "the probability of event \(A\)"
The likelihood of an event \(A\) occuring is measured on a scale that goes from \(0\) to \(1\), where:
The following illustration gives us an idea of how probabilities are measured:
All probabilities fit somewhere on this scale (no exceptions), where:
Given an event \(A\), the following result will always be true: \[0 \leq p\begin{pmatrix}A\end{pmatrix} \leq 1\] Note: this result is important, make a note of it. If ever, in your calculations, you find a probability greater than \(1\), or less than \(0\), you have done something wrong. Go back and check your working.